Question

A is a set containing n elements. Two subsets P and Q of A are chosen at random. (P and Q may have elements in common). The probability that $P\cup Q\ne A$ is

• A. $(3/4{)}^n$
• B. $1/4n$
• C. ${}^n{C}_2/2^n$
• D. $1-(3/4{)}^n$

Answer 1

Answer: (D)

$1-(3/4{)}^n$

Let A $=\{x_{1,}{x}_2,...x_n\}$.
Each $x_i\in A$ has four choices
i) $x_i\in P$ and $x_i\in Q$
ii) $x_i\in P$ and $x_i\notin Q$
iii) $x_i\notin P$ and $x_i\in Q$
iv) $x_i\notin P$ and $x_i\notin Q$
For $P\cup Q=A$ case (iv) is not favourable.
probability $=\frac{3^n}{4^n}$
Hence required probability $=1-\frac{3^n}{4^n}$